Page:BatemanElectrodynamical.djvu/30

 Putting $$dx=\delta x$$ and extracting the square root, we obtain an invariant

Now, let

$B_{x}dy\ dz+B_{y}dz\ dx+B_{z}dx\ dy+E_{x}dx\ dt+E_{y}dy\ dt+E_{z}dy\ dt,$

be an invariant of the second order. Multiplying it by (12) and rejecting a factor $$\sqrt{\Theta}dx\ dy\ dz\ dt$$, we obtain an invariant

$D_{x}\delta y\ \delta z+D_{y}\delta z\ \delta x+D_{z}\delta x\ \delta y-H_{x}\delta x\ \delta t-H_{y}\delta y\ \delta t-H_{z}\delta z\ \delta t,$

where

$\begin{array}{c} \sqrt{\Theta}D_{x}=\kappa_{11}E_{x}+\kappa_{12}E_{y}+\kappa_{13}E_{z}+\kappa_{14}B_{x}+\kappa_{15}B_{y}+\kappa_{16}B_{z},\\ \dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\dots\qquad\\ -\sqrt{\Theta}H_{x}=\kappa_{41}E_{x}+\kappa_{42}E_{y}+\kappa_{43}E_{z}+\kappa_{44}B_{x}+\kappa_{45}B_{y}+\kappa_{46}B_{z}.\end{array}$

The relation between the two invariants will be a mutual one if the coefficients $$\kappa_{11},\ \dots$$. are the elements of an orthogonal matrix.

The Invariants of a Spherical Wave Transformation.
Starting from the fundamental invariants

we may obtain a number of others by the methods of multiplication and reciprocation. It will be sufficient to enumerate these if we mention the equations from which they are derived,

$\begin{array}{ccc} (1\ \mathsf{and}\ 6) & \frac{1}{\lambda^{2}}\left[\rho w_{x}dx+\rho w_{y}dy+\rho w_{z}dz-\rho dt\right], & (7)\\ \\(5\ \mathsf{and}\ 4) & \left(E_{x}^{2}+E_{y}^{2}+E_{z}^{2}-H_{x}^{2}-H_{y}^{2}-H_{z}^{2}\right)dx\ dy\ dz\ dt & (8)\\ \\(4\ \mathsf{and}\ 4) & \left(E_{x}H_{x}+E_{y}H_{y}+E_{z}H_{z}\right)dx\ dy\ dz\ dt & (9)\\ \\(3\ \mathsf{and}\ 6) & \rho\left[A_{x}w_{x}+A_{y}w_{y}+A_{z}w_{z}-\Phi\right]dx\ dy\ dz\ dt & (10)\\ \\(6\ \mathsf{and}\ 7) & \frac{\rho^{2}}{\lambda^{2}}\left(1-w^{2}\right)dx\ dy\ dz\ dt, & (11)\\ \\(5\ \mathsf{and}\ 7) & \frac{\rho}{\lambda^{2}}\left[\left(E_{x}-w_{z}H_{y}+w_{y}H_{z}\right)dy\ dz\ dt+\left(E_{y}-w_{x}H_{z}+w_{z}H_{x}\right)dz\ dx\ dt\right. & (12)\\ & \left.+\left(E_{z}-w_{y}H_{z}+w_{x}H_{y}\right)dx\ dy\ dt-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)dx\ dy\ dz\right],\\ \\(12) & \frac{\rho}{\lambda^{4}}\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right. & (13)\\ & \left.+\left(E_{z}+w_{z}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right],\\ \\(13\ \mathsf{and}\ 2) & \rho\ dx\ dy\ dz\ dt\left[\left(E_{x}+w_{y}H_{z}-w_{z}H_{y}\right)\delta x+\left(E_{y}+w_{z}H_{x}-w_{x}H_{z}\right)\delta y\right. & (14)\\ & \left.+\left(E_{z}+w_{x}H_{y}-w_{y}H_{x}\right)\delta z-\left(w_{x}E_{x}+w_{y}E_{y}+w_{z}E_{z}\right)\delta t\right].\end{array}$|undefined